The 28-year-old, whose list of high-seeded victims at previous Australian Opens includes Ana Ivanovic (19th), Serena Williams (12th), Angelique Kerber (5th) and Simona Halep (3rd), will now face either Johanna Konta or Caroline Wozniacki in the fourth round.
Makarova, who despite her giant-killing exploits has only reached one semi-final at Melbourne Park (2015), raced through the first set courtesy of three breaks and then into a 4-0 lead in the second.
Cibulkova, however, fought back to win the next five games to take a 5-4 lead then went on to force a deciding set.
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Makarova took a medical time out for treatment on her right elbow while leading 3-2 in the third but it did not see the momentum swing back Cibulkova. The Russian then broke to take a 5-3 lead and served out in the next game.
(Writing by Greg Stutchbury in Wellington; Editing by Peter Rutherford)
